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Unformatted text preview: Chapter 7. Functions of Several Variables 7.1 Introduction In elementary calculus, we encountered scalar func- tions of one variable, e.g. f = f ( x ). However, many physical quantities in engineering and science are described in terms of scalar functions of several variables. For example, (i) Mass density of a lamina can be described by δ ( x,y ), where ( x,y ) are the coordinates of a point on the lamina. (ii) Pressure in the atmosphere can be described by P ( x,y,z ) where ( x,y,z ) are the coordinates of a point in the atmosphere. 2 MA1505 Chapter 7. Functions of Several Variables (iii) Temperature distribution of a heated metal ball can be described by T ( x,y,z,t ) where ( x,y,z ) are the coordinates of a point in the ball and t is the time. 7.1.1 Functions of Two Variables A function f of two variables is a rule that assigns to each ordered pair of real numbers ( x,y ) a real number denoted by f ( x,y ). We usually write z = f ( x,y ) to indicate that z is a function of x and y . Moreover, x,y are called the independent variables and z is called the dependent variable. The set of all ordered pairs ( x,y ) such that f ( x,y ) can be defined is called the domain of f . 2 3 MA1505 Chapter 7. Functions of Several Variables 7.1.2 Example (a) f ( x,y ) = x 2 y 3 . This is a function of two variables which is defined for any x and y . So the domain of f is the set of all ( x,y ) with x,y ∈ R . (b) f ( x,y ) = p 1- x 2- y 2 . This function is only defined when 1- x 2- y 2 ≥ 0, or equivalently x 2 + y 2 ≤ 1. So the domain of f is the set D = { ( x,y ) : x 2 + y 2 ≤ 1 } . Note that D represents all the points in the xy plane lying within (and on) the unit circle. 3 4 MA1505 Chapter 7. Functions of Several Variables (c) We can also define f in “pieces” as a compound function . For example f ( x,y ) =    √ x- y if x > y, √ y- x if x < y, 1 if x = y . 7.1.3 Functions of Three or More Variables We can define functions of three variables f ( x,y,z ), four variables f ( x,y,z,w ), etc in a similar way. 7.2 Geometric Representation 7.2.1 Graphs of functions of two variables The graph of a function f ( x ) of one variable is a curve in the xy-plane, which can be regarded as the set of all points ( x,y ) in the xy-plane such that y = f ( x ) . By analogy, we have the graph of a function f ( x,y ) 4 5 MA1505 Chapter 7. Functions of Several Variables of two variables is the set of all points ( x,y,z ) in the three dimensional xyz-space such that z = f ( x,y ) . This set represents a surface in the xyz-space. 7.2.2 Example The graph of f ( x,y ) = 5- 3 x- 2 y is the plane with equation z = 5- 3 x- 2 y (or 3 x + 2 y + z = 5). 7.2.3 Example The graph of g ( x,y ) = 8 x 2 + 2 y 2 is the paraboloid (see diagram below) with equation z = 8 x 2 + 2 y 2 ....
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